Why are solenoidal fields called solenoidal? A solenoidal tangent field, mathematically speaking, is one whose divergence vanishes. They are also called incompressible. I understand why they are called incompressible — a fluid flow is called incompressible when a small fluid parcel retains constant density when it moves along along a streak line. This means that its material derivative vanishes and this in turn means that the divergence of its velocity field vanishes.
But why the description of such a field as solenoidal? I expect that this name had historical origins but it's unlikely that it was so named without some link to some aspect of what is generally meant by a solenoid. However, checking a few sources online hasn't resolved this mystery. Generally, a solenoidal field is defined mathematically as above without any discussion why it was named as such.
Now, a solenoid is a helix which suggests that a fluid particle whose velocity field is solenoidal should be moving helically. Is this because such a field would normally be rotational and so has a non-zero curl and so fluid particles in small tubes around a streakline will move helically and because the motion is also incompressible, the cross-section of this tube is constant and hence we have described a solenoid, of a kind?
 A: [To expand on Wojowu's comment.]
Q: "Why the description of a divergence-free field as solenoidal? I expect that this name had historical origins but its unlikely that it was so named without some link to some aspect of what is generally meant by a solenoid."A: The name solenoid for a helical coil was invented by Ampère (1823), who is quoted as follows in Wikipedia:

l'assemblage de tous les circuits qui l'entourent,
assemblage auquel j'ai donné le nom de solénoïde électro-dynamique, du
mot grec σωληνοειδὴς, dont la signification exprime précisement ce qui
a la forme d'un canal, c'est-à-dire la surface de cette forme sur
laquelle se trouvent tous les circuits.

the assembly of all the circuits around it, to which I have given the name of electro-dynamic solenoid, from the Greek word σωληνοειδὴς, the meaning of which expresses precisely what has the shape of a channel, that is to say the surface of this form on which all the circuits are located.
The magnetic field lines created by a solenoid are divergence free, which motivates the general name "solenoidal" for a divergence free field; it might be possible to locate some early incidences of this use, but the link to Ampère's solenoid seems beyond debate.

